Année 2017

JANVIER

Mardi 10, Salle 01 à 10h

Jean Marc Bardet (Paris 1, SAMM) Title: Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process

We first provides some key-results about the parametric and semi-parametric estimation of the Hurst parameter of long-memory processes. Then, we consider the particular case of the estimation of the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations.

We consider the Voronoi tessellation based on a homogeneous Poisson point process in n R^d. For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson-Delaunay tessellation is also discussed.

MARS

Our main objective is to describe the large deviation behavior and limiting path properties for matrix recursive sequences. Motivated by branching processes in random environments, matrix recursions were originally introduced in the seminal paper of Kesten (1973), who studied the recursive sequence V(n) = A(n) V(n-1) + B(n), where {(A(n)} is an i.i.d. sequence of random matrices and {B(n)} an i.i.d. sequence of random vectors. Under a stationarity condition, Kesten showed that V(n) converges to a random variable V, whose tail decays at a specified polynomial rate. More recent work has applied this estimate in a variety of other areas, such as financial time series modeling.

In the first part of the talk, working under Kesten’s assumptions, we derive an extremal estimate for the first passage time and exhibit that, under a large excursion, the empirical average converges to an exponentially-shifted Markov random walk. Next, we turn to the case where the process {V(n)} is explosive, and derive a related large deviation estimate and certain conditioned limit theorems. (Based on collaborations with Sebastian Mentemeier and Anand Vidyashankar.)

AVRIL

Mardi 25 Avril, Salle 314 à 10h

Konstantinos Fokianos (Cyprus)

Title: Testing independence for multivariate time series by the auto-distance correlation matrix

We introduce the notions of multivariate auto-distance covariance and correlation functions for time series analysis.

These concepts have been recently discussed in the context of both independent and dependent data but we extend them in a different direction by putting forward their matrix version.

Their matrix version allows us to identify possible interrelationships among the components of a multivariate time series. Interpretation and consistent estimators of these new concepts are discussed. Additionally, we develop a test for testing the hypothesis of i.i.d. for multivariate time series data. The resulting test statistic performs better than the standard multivariate Ljung-Box test statistic. This is joint work with M. Pitsillou.

Abstract: The Principle of Conditioning is a heuristic rule that allows transferringlimit theorems for independent random variables into limit theorems for depen-dent random variables. While the limit theorems obtained via the Principle ofConditioning are commonly known, methods of verifying their assumptions arealways of interest due to potential applications.Our motivation comes from the work by Jara, Komorowski and Olla (2009),where the functional limit theorems due to Durret and Resnick (1978) wereused to obtain a fractional di_usion as a scaled limit of solutions of a linearBoltzmann equation (with Markov chains providing probabilistic solutions ofthe latter).We have obtained a di_erent set of conditions ensuring that partial sums ofa stationary Markov chain converge to a stable law with exponent _ 2 (0; 2).The conditions are related to operator properties of the transition probabilities.The approach is through the \Main Lemma" of the Principle of Conditioning.This is a joint work with Mohamed El Machkouri and Dalibor Voln_

Mardi 22, Salle 314 à 10h

Ivan Nourdin "Phase Singularities in complex arithmetic Random Waves"

résumé: "Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We will use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result will be that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. The talk will be based on a joint work with Federico Dalmao, Giovanni Peccati and Maurizia Rossi. »

Abstract: In this paper we study a local polynomial estimator of the regression function and its derivatives. We propose a sequential technique based on a multivariate counterpart of the stochastic approximation method for successive experiments for the local polynomial estimation problem. We present our results in a more general context by considering the weakly dependent sequence of stream data, for which we provide an asymptotic bias-variance decomposition of the considered estimator. Additionally, we study the asymptotic normality of the estimator and we provide algorithms for the practical use of the method in data streams framework.

OCTOBRE

We study the estimation risk induced by univariate and multivariate methods for evaluating the conditional Value-at-Risk (VaR) of a portfolio

of assets. The composition of the portfolio can be time-varying and the individual returns are assumed to follow a general multivariate

dynamic model. Under ellipticity of the conditional distribution, we introduce in the multivariate framework a concept of VaR parameter,

and we establish the asymptotic distribution of its estimator. A multivariate Filtered Historical Simulation method, which does not rely on ellipticity, is studied.

We also consider two univariate approaches based on past real or reconstituted returns. We derive asymptotic confidence intervals for the conditional VaR, which allow to quantify simultaneously the market and estimation risks. Potential usefulness, feasibility and drawbacks of the different univariate and multivariate approaches are illustrated via Monte-Carlo experiments and an empirical study based on stock returns.

Multi-state models are generally used for pricing and reserving long-term care (LTC) insurance contracts. While most of the current researches assume that the model is Markovian, we show in this paper that this assumption should actually be rejected, as it leads to a bias in the estimation procedure that may be significant. Since the transition probabilities are complex to estimate with an inhomogeneous semi-Markov model based on transition intensities, we choose to apply recent methods for a direct estimation of transition probabilities, which perform better than the Aalen-Johansen estimator when the Markov assumption is not satisfied. Using the so-called pseudo-values related to Jackknife methods on these estimators, we incorporate the effects of covariates (duration, sex and generation) with a GLM regression model, similarly to Helms et al. [Helms, F., Czado, C. and Gschlößl, S. Calculation of Premiums LTC based on Direct Estimates of transition probabilities. Astin Bulletin, 2005, 455-469]. Another key interest of our approach is to include the diseases which cause the entry into dependency as it affects strongly the residual lifetime of LTC claimants. We apply it on a real French LTC insurance portfolio and analyze the effect of Markov hypothesis both on pricing and on the solvency capital requirement calculation.

Mardi 27, Salle 201 à 10h

I. Grublite (Cergy & Vilnius) TBA

JUIN

Abstract: In this paper we consider the projection of mortality surface at the national level. We consider modeling mortality improvements on a cohort basis taking into account correlations across generations. Therefore, we propose to model the whole mortality surface by considering a random field approach with a specific causal structure instead of a univariate modeling framework. Such an approach has the advantage to account for a local dependence among adjacent cohorts.

MAI

Mardi 31, salle 421 à 10h

Mathieu Rosenbaum (Paris 6) Nearly unstable Hawkes processes.

Abstract: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this talk such processes for which the stability condition is almost violated, focusing in particular on limit theorems.

Mardi 24, salle 201 à 15h

Jacek Leśkow (Cracow University of Technology) A class of nonstationary, periodically or almost periodically correlated time series that are weakly dependent.

Abstract: The focus will be on applications in telecommunication and mechanical signal processing. In a relatively simple model we will show how to study the asymptotic properties of the estimators. We will also introduce the concept of resampling and investigate some aspects of consistency of selected resampling procedures.

AVRIL

Abstract: The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for Gaussian mixtures. We study the case of mixtures of autoregressive models (i.e. independent regime switches). In this framework, we give sufficient conditions to keep the LRTS tight and compute its asymptotic distribution. Some numerical examples illustrate the results and their convergence properties.